I have noticed in some Shuls the Congregants repeat the counting of the Omer by day time (without a Brocha). What is the source to do this? Is this done as a precaution just in case one forgot to count the previous night, or is there a reason to recount by day?
The mitzvah of counting Omer with a blessing is only at night, as explained here. If one forgets, he may count the next day without a blessing.
Thus, it is as you surmised. The congregation is doing it as a reminder. It would help, perhaps, if you stated what type of shul, congregation you attend. I'm guessing that it is not Orthodox. No Orthodox shul that I have attended does this congregationally, as many of their people have counted at night either by attending the minyan, or they assume that people have done it. Regardless, most shuls will have the Gabbai announce after davening, "Last night we counted xx", and that reminds someone who forgot to say it, himself. I.e., no Orthodox place I know of does a congregational counting during the day.
It's possible that your shul may have less or non-educated people or travelers / visitors attending Shacharit who don't know the bracha and the rules that well. You're rav may be trying to cater to their needs.